Define capacitance of a parallel plate capacitor,Derive an expression for the capacitance of a parallel plate capacitor with dielectric of thickness "t," where "(t

Explanation

Capacitance of a parallel plate capacitor with vacuum between the two plates is
`C_(0)=(epsi_(0)A)/(d)`
Let a plate A be given +Q charge and plate B be given -Q charge. Introduce dielectric slab, say of thickness t(t If `vecE_(0)` is the total field between A and B, the polarisation in the dielectric introduces a field `vecE_(p)` in the opposite direction.
`therefore`net electric field inside the dielectric is
`vecE=vecE_(0)-vecE_(p)`.
Potential difference between A and B is

But `(E_0)/(E)=K`, the dielectric constant.
`thereforeE=(E_(0))/(K)`
or `V=E_(0)(d-t)+(E_(0))/(K)xxt`
or `V=E_(0)[d-t+(t)/(K)]`
But `E_(0)=(sigma)/(epsi_(0))=(Q)/(Aepsi_(0))`
`therefore V=(Q)/(Aepsi_(0))[d-t+(t)/(K)]`
But capacity, `C=(Q)/(V)`
`therefore C=(Q)/((Q)/(Aepsi_(0))[d-t+(t)/(K)])`
or `Q=(Aepsi_(0))/(d-t+(t)/(K))`
If whole of the space between plates is filled with dielectric, then t=d
`therefore C=(Aepsi_(0))/(d-d+(d)/(K))`
or `C=(KAepsi_(0))/(d)`.